Numerical analysis of finite-time blow-up in nonlinear integro-differential equations

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Nonlinear integral equations of Volterra type arise in the mathematical modelling of thermal ignition in solid combustible materials (e.g. in the formation of shear bands in steel, when subjected to very high strain rates; see the review paper by C.A. Roberts in J. Comput. Appl. Math 205 (2007), 736-743, and its references). Such equations -- whose solutions blow up in finite time -- are also related to semilinear parabolic partial differential equations of reaction-diffusion type; compare the book 'Mathematical Problems from Combustion Theory' (1989) by J. Bebernes and D. Eberly, and the review paper by H.A. Levine in SIAM Review 32 (1990), 262-288. In this proposal we shall study the numerical analysis and computational solution of nonlinear Volterra equations with blow-up solutions, with the focus being on computing the blow-up time and deriving realistic error estimates. Since in many applications it is not known a priori if the solution will blow up in finite time or if it exists globally, we shall in particular address the important problem of detecting blow-up numerically. This will be done both for nonlinear Volterra integral equations and for semilinear parabolic integro-differential equations (containing nonlocal reaction terms in the form of Volterra memory terms)). In te latter case the theory of finite-time blow-up when the underlying spatial domain is unbounded (e.g. a sectorial or cone-like domain) is essentially open. The proposal will also try to close this gap, since a thorough understanding of this theory (e.g. the location of the blow-up points) will be the key for designing effective numerical schemes for such nonlocal blow-up problems.